Do I need a visa to speak in a conference in UK?

This depends a lot on what the exact circumstances of "speak in a symposium" are. This brief description can match both completely unproblematic situations and situations that will get you into trouble.

• The unproblematic case: You're attending an academic conference, funded by a grant from the conference organizers. At some point during the conference you're going to stand in front of a crowd of all or some of the attendees for somewhere between 15 and 30 minutes and present some of your research -- just like 80 % of people who attend academic conference do.

• The asking-for-trouble case: You're a star-billed speaker at the conference, and attendees are paying for separate tickets to hear Lucas P teach them everything about widgetology. Even if you're not earning cash from the gig, you do get a first-class vacation out of it, nice hotel rooms, expensive dining, etc -- and the organizers hope to make a profit from having you there.

In the first case you will be an ordinary visitor for business, and should not encounter major problems. Have documentation for your source of funding, contact information for the funders and so forth ready when you approach the immigration desk -- but don't present this documentation except when asked for it; probably you won't be.

In the second case, what you're proposing to do will probably count as working, and you would likely be denied entry and removed from port.

Somewhere in between? You, and later the immigration officer, will need to determine whether it is most like the easy case or most like the troublesome case. Some variation from the completely vanilla case could still be okay (such as: "I'm an invited plenary speaker, so I have 60 minutes instead of 30, and my name appears higher on the conference website than ordinary paper authors") -- but if you're in doubt and have the time, it would be prudent to apply for entry clearance (i.e. a "visa") from the UK consulate in advance. Having entry clearance will not allow you to do anything you wouldn't be allowed to without it, but since your story will have been checked in advance, the risk of being rejected at the border decreases dramatically.

If you do go without a visa, it is important to remember to describe your purpose of travel as attending a conference rather than speaking at a conference. You may not attach much importance to the distinction, but the latter is much more likely to send alarm bells ringing for the immigration officer. Even if you can clear it up eventually, it could be the difference between "very well, stampstampstamp welcome to the UK" and several hours of questioning.